# Properties

 Label 714f Number of curves $4$ Conductor $714$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("714.e1")

sage: E.isogeny_class()

## Elliptic curves in class 714f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
714.e4 714f1 [1, 1, 1, 1, 101] [4] 192 $$\Gamma_0(N)$$-optimal
714.e3 714f2 [1, 1, 1, -319, 2021] [2, 2] 384
714.e2 714f3 [1, 1, 1, -679, -3883] [2] 768
714.e1 714f4 [1, 1, 1, -5079, 137205] [2] 768

## Rank

sage: E.rank()

The elliptic curves in class 714f have rank $$1$$.

## Modular form714.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - q^{12} - 6q^{13} - q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.